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Resource-efficient digital quantum simulation of $d$-level systems for photonic, vibrational, and spin-$s$ Hamiltonians
Nicolas P. D. Sawaya, Tim Menke, Thi Ha Kyaw, Sonika Johri, Alán Aspuru-Guzik, Gian Giacomo Guerreschi
TL;DR
The paper addresses how to encode d-level quantum systems into qubits when prior simulation work has focused mainly on binary degrees of freedom. It develops a general encoding and resource-counting framework, finding that Gray-code properties and conversions between encodings can reduce costs, while the best choice depends strongly on the application and d. The analysis is scoped by assumptions including all-to-all qubit connectivity.
Problem
Which integer-to-bit encodings are most efficient for d-level quantum simulations on hardware constrained by qubit and operation resources?
Method
The paper maps d-level operators to Pauli strings, analyzes Hamming distance and local-operator costs, and compares composite Hamiltonian strategies with optional encoding conversions.
Results
Optimal encoding depends strongly on Hamiltonian class and d; Gray code tends to favor tridiagonal operators, standard binary common diagonal operators, and conversions often reduce total operations.
Takeaways & Limitations
Encoding analysis can guide qubit-based simulation of bosonic, spin-s, vibrational, molecular, and other d-level subsystems under hardware resource constraints.
Takeaways & Limitations
The resource counts assume all-to-all qubit connectivity; hardware topology and SWAP gates may modify the counts and observed trends.
Abstract
from arXiv · showhide
Simulation of quantum systems is expected to be one of the most important applications of quantum computing, with much of the theoretical work so far having focused on fermionic and spin-$\frac{1}{2}$ systems. Here, we instead consider encodings of $d$-level (i.e. qudit) quantum operators into multi-qubit operators, studying resource requirements for approximating operator exponentials by Trotterization. We primarily focus on spin-$s$ and truncated bosonic operators in second quantization, observing desirable properties for approaches based on the Gray code, which to our knowledge has not been used in this context previously. After outlining a methodology for implementing an arbitrary encoding, we investigate the interplay between Hamming distances, sparsity patterns, bosonic truncation, and other properties of local operators. Finally, we obtain resource counts for five common Hamiltonian classes used in physics and chemistry, while modeling the possibility of converting between encodings within a Trotter step. The most efficient encoding choice is heavily dependent on the application and highly sensitive to $d$, although clear trends are present. These operation count reductions are relevant for running algorithms on near-term quantum hardware because the savings effectively decrease the required circuit depth. Results and procedures outlined in this work may be useful for simulating a broad class of Hamiltonians on qubit-based digital quantum computers.
I. INTRODUCTION
This section motivates encoding d-level systems into qubits and frames encoding choice as a hardware-budget problem. It introduces several encodings and reports that optimal choices vary with Hamiltonian class and d.
- Many quantum-physics problems involve d-level particles, including bosons, vibrational modes, spin-s particles, and molecular energy levels.
- The mapping of d-level systems to qubits remains underexplored, particularly regarding which encodings are optimal for specific operators, problem instances, and hardware.
- Hardware budgets constrain both qubit counts and gate counts, so an encoding suitable for one device may be unsuitable for another.
- Across five Hamiltonian classes, optimal encoding strategies vary with Hamiltonian type and d, sometimes favoring conversions between encodings and sometimes unary storage.
- The study analyzes unary, standard binary, Gray, and block unary encodings, including arbitrary d-level-to-qubit mappings and conversion circuits.
- Encoding definitions: Standard binary uses Nq = ⌈log2 d⌉ qubits, whereas unary uses Nq = d qubits and occupies only an exponentially small Hilbert-space subspace.
- Encoding definitions: Gray code assigns adjacent integers bit strings with Hamming distance one, making it favorable for tridiagonal operators with zero diagonals.
B. Mapping d-level Matrix Operators to Qubits
The paper maps d-level matrix elements to Pauli-string sums through arbitrary integer-to-bit encodings. It then links Hamming distance to Pauli-string complexity and entangling-gate requirements.
- A d-level matrix operator is decomposed into terms |l⟩⟨l′|, whose integer labels are mapped to qubit bit strings using an arbitrary encoding.
- For non-compact encodings, the mapping can ignore qubits outside the bitmask union associated with the two basis states.
- Each qubit-local transition is converted into Pauli operators, producing a sum of Pauli strings for every encoded matrix element.
- Hamming distance: Hamming distance H(l,l′) counts unequal bits between encoded states, and larger distances increase the average Pauli-string length.
- Hamming distance: Smaller Hamming distance reduces non-identity Pauli operators, which lowers the operations needed to implement the corresponding Hamiltonian terms.
- Hamming distance: The Gray code has fewer Pauli operators per Pauli string than the compared alternative in the discussed example, enabling fewer operations.
- Circuit implementation: Exponentiating a Pauli string with p Pauli operators requires 2(p −1) two-qubit gates before considering cancellations across a Trotter product.
D. Avoiding superfluous terms in non-compact codes
For non-compact encodings, multiply local operators in the matrix representation before mapping them to qubits to avoid superfluous Pauli terms. Resource estimates then depend on encoding, Hamming distance, sparsity, and circuit cancellation, so asymptotic bounds should guide rather than replace numerical comparisons.
- Avoiding superfluous terms: Matrix multiplication before qubit encoding avoids superfluous terms that can arise when already encoded operators are multiplied.This is especially relevant for squared operators, where the matrix representation of A^2 should be encoded directly rather than obtained by squaring the encoded A.
- Asymptotic resource bounds: Compact encodings have O(d2 log d) upper bounds for both tridiagonal and dense operators, while unary has O(d2) for dense operators.As matrix density increases, the comparative advantage of unary decreases, although commonly used O(d)-density operators may still favor unary gate counts.
- Hamming distance and sparsity: For tridiagonal operators, Gray encoding has lower upper bounds because every nonzero term has unit Hamming distance.The relevant sparsity pattern includes canonical bosonic position and momentum operators and spin-s operators.
- Optimization and practical counts: For smaller d, asymptotic bounds are mainly trend indicators, so numerical testing of all encodings is preferable for identifying the fewest operations.The resource analysis targets a single Suzuki-Trotter step and uses weighted Pauli-string decompositions with term combination.
- Asymptotic resource bounds: Unary encoding keeps the relevant bitmask size constant, whereas Gray and standard binary require K = ⌈log2 d⌉, explaining lower asymptotic bounds for unary in the tridiagonal case.Block unary with g = 3 has intermediate K and intermediate upper bounds.
F. Diagonal Binary-Decomposable Operators
Diagonal binary-decomposable (DBD) operators can have especially efficient standard-binary implementations, particularly when d is a power of two. Truncation choices can therefore affect entangling-gate requirements.
- DBD operators are diagonal operators whose entries can be expressed through binary-decomposable terms.
- When log2 d is an integer, exp(−iθn̂) requires only log2 d single-qubit rotations and no entangling gates under standard binary.
- For spin-s Sz, the standard-binary advantage occurs when d = 2s + 1 is a power of two.
- For truncated bosonic n̂ with a minimum d = 11, using d = 16 can be more efficient than using the smaller truncation.
- Standard binary is superior for the illustrated DBD operator at d = 4, while d = 3 requires entangling gates; squaring generally removes the DBD property.
G. Local operators
The local operators studied have varied sparsity patterns, and encoding efficiency depends on how those patterns interact with Hamming distances, truncation, and diagonal structure. Gray, standard binary, and unary each become favorable in different cases.
- Bosonic ladder operators connect only neighboring levels, producing tridiagonal position and momentum operators.
- The square of q̂ is pentadiagonal with nonzero off-diagonal entries separated by two levels, making Gray’s unit Hamming distance less useful.
- For q̂2, Gray outperforms standard binary asymptotically, while standard binary is better at lower d; a gate-count dip occurs at d = 8 because the diagonal is DBD.
- Increasing bosonic truncation from d = 5 to d = 8 can reduce gates without increasing qubit count, but this is unavailable for spin-s systems because it permits unphysical states.
- Unary is inferior in both qubit and gate counts for several low-dimensional bosonic operators, including q̂ at d = 4, 7, and 8.
- For spin operators, standard binary has no entangling gates for Sz at d = 4 and 8, while Gray is usually better for tridiagonal Sx and unary is often inferior.
- Block unary with g = 3 is never the sole lowest-entangling-gate encoding and would matter only under limited hardware budgets.
- The reported savings are generally constant-factor because d usually does not grow with particle number.
H. Conversions between encodings
Encoding conversions allow calculations to use different representations for different operators while managing memory. The paper gives general conversion circuits and includes their costs in composite resource estimates.
- Different Bose-Hubbard terms favor different encodings: n̂ is usually more efficient in standard binary, whereas hopping is usually more efficient in Gray.
- Conversion costs can be substantially smaller than the resource-efficiency gap between encodings, making repeated compaction and expansion advantageous.
- Gray-to-standard-binary conversion uses (⌈log2 d⌉−1) sequential CNOT gates.
- The paper provides circuits for standard-binary–unary conversion at arbitrary truncation d, with modifications when d is not a power of two.
- Unary conversion can be relevant when extra quantum resources are available, but its cost becomes significant as d increases.
- Standard-binary–unary conversion resources are tabulated for arbitrary d, while unary-to-standard-binary conversion uses the reverse procedure.
- Block-unary conversions are not further analyzed because block unary is expected to have limited utility and decoherence may prevent conversion.
I. Composite Systems
Composite Hamiltonian costs depend strongly on the Hamiltonian, d, qubit budget, and whether conversions are allowed. The optimal strategy ranges from staying compact to using unary throughout or compacting around unary operations.
- The study compares five encoding schemes using optimized Trotter circuits and inter-encoding conversion costs for five physical and chemical Hamiltonian classes.
- The best encoding can change sharply with d, so the paper highlights d = 4 and d = 10 and reports additional results through d = 16.
- Scenario A: Scenario A uses one compact encoding throughout, keeping each particle within log2 d qubits; examples include Bose-Hubbard and 1D QHO at d = 4.
- Scenario B: Scenario B allows standard-binary–Gray conversion while retaining compact memory, as for Heisenberg s = 7/2 and Franck-Condon d = 4.
- Scenario C: Scenario C uses unary for operations but repeatedly compacts to standard binary or Gray when saving memory remains cheaper than staying unary.
- Scenario D: Scenario D keeps data in unary throughout when qubit space permits; otherwise, it is better to perform all operations in Gray and/or standard binary.
- Memory-efficient schemes reduced two-qubit entangling-gate counts by up to 33% versus unary-only and up to 49% versus standard-binary-only.
III. DISCUSSION
The study shows that encoding efficiency depends strongly on operator structure, Hamiltonian class, and truncation level, with encoding conversion often improving resource use. However, hardware connectivity and resource metrics can alter the observed trends.
- Gray tends to outperform SB for tridiagonal operators, whereas SB tends to outperform Gray for a common diagonal operator class.
- Converting between encodings within a Suzuki-Trotter step often reduces operation counts.
- Unary can reduce operations but requires more qubits, while SB or Gray often minimize both qubit and operation counts.
- The optimal encoding and conversion strategy varies substantially across five Hamiltonian classes and truncation levels d.
- Resource estimates assume all-to-all connectivity, so hardware topology and SWAP gates may modify the counts and observed trends.
- The methodology is intended to support resource-efficient simulation of physically and chemically relevant Hamiltonians.
FIGURE LEGENDS
The figure legends describe resource comparisons, circuit constructions, asymptotic bounds, and encoding-conversion schemes for qubit-based Hamiltonian simulation. They emphasize that gate counts depend on operator structure, encoding, and circuit optimization.
- Figure 2 compares two-qubit gate costs for five Hamiltonians relative to standard binary under first-order Trotterization.
- Exponentiating a Pauli string requires 2(p −1) two-qubit gates, while adjacent Trotterized exponentials can exhibit substantial gate cancellation.
- For tridiagonal operators, SB and Gray use logarithmically many qubits, while unary and block unary use linearly many qubits asymptotically.
- Figure 8 illustrates expanding to unary for an operator and then compacting back to SB or Gray when memory resources permit.
I. UPPER BOUNDS FOR ENTANGLING GATES
The upper-bound analysis links entangling-gate requirements to encoding-dependent Hamming distances, Pauli-string lengths, and operator sparsity. Unary has lower bounds for sparse operators but uses more qubits, while compact-code advantages emerge as density increases.
- For Hermitian level pairs, Pauli-string length distributions depend on the Hamming distance between their encoded bit strings.
- For sparse tridiagonal structure, compact Gray and SB encodings have an O(d^2 log d) entangling-gate upper bound.
- For the same sparse structure, unary has an O(d) upper bound because each term pair requires a d-independent number of CNOT gates.
- The unary advantage comes at the cost of requiring more qubits than compact encodings.
- For dense operators, compact encodings have an O(d^2 log d) absolute entangling-gate limit, reducing unary’s asymptotic advantage.
- Analytical bounds explain general trends but may overestimate optimized circuits because repeated Pauli terms combine and gates can cancel.
II. CIRCUIT OPTIMIZATION
Circuit resources are estimated by Pauli decomposition, CNOT-ladder synthesis, and repeated optimization passes that cancel or merge gates. Ordering affects cancellation, while conversion circuits extend the set of usable encodings but add costs.
- Each d-level operator is converted to Pauli terms, whose coefficients are collected and whose exponentials are synthesized as CNOT ladders.
- The optimizer cancels inverse gates, merges rotations, commutes gates, and repeats pattern searches until the circuit converges.
- Pauli-term ordering changes gate counts because neighboring CNOT ladders determine which gates can cancel.
- The default ordering outperformed more than 900 random orderings tested for each of the unary, Gray, and SB encodings of q-hat.
- SB-to-unary conversion requires modifications when d is not a power of two, including changing controls and removing unnecessary qubits.
- Conversion between SB and block-unary encodings is possible, but its cost may make block unary unsuitable under tight gate budgets.
V. FIRST QUANTIZATION
First-quantized operators can represent bosonic degrees of freedom and support position–momentum basis changes through the quantum Fourier transform. The approach reduces some operator costs, but initial-state preparation remains a significant hurdle.
- A Trotter step can exponentiate position and momentum squares by alternating basis-diagonal operations with QFT and QFT^-1.
- Trotterizing first-quantized position and momentum operators requires log2 d single-qubit rotations and zero two-qubit gates.Exponentiating their squares still requires entangling gates.
- The QFT contributes O(N^2 log2 d) operations in the described implementation.
- Initial-state preparation is a significant computational hurdle for first quantization.Preparing a QHO ground state requires preparing a specific nontrivial state.
VI. HAMILTONIAN DEFINITIONS
The paper defines representative bosonic, spin, optical, and molecular Hamiltonians to evaluate encoding resources across several physics and chemistry applications. These models differ in truncation, particle or mode counts, and structural assumptions such as sparse molecular couplings.
- Bose-Hubbard model: The Bose-Hubbard model describes bosonic modes on discrete lattice sites and includes Mott-insulator and superfluid phases.The paper studies a one-dimensional periodic version parameterized by levels d per site and particle number N.
- Shifted one-dimensional QHO: The shifted one-dimensional QHO models a bosonic system and is directly relevant to molecular Franck-Condon factors.Its displacement δ influences how many levels d have appreciable intensity.
- Multidimensional Franck-Condon factors: Multidimensional Franck-Condon calculations require transitions across vibrational Fock states and use a Duschinsky matrix to transform between potential-energy-surface bases.
- Multidimensional Franck-Condon factors: The molecular simulations assume a constant number k=4 of nonzero entries per Duschinsky-matrix row and the same level count d for every vibrational mode.These approximations support comparisons of typical constant-factor savings between mappings.
- Spin-s Heisenberg model: The transverse spin-s Heisenberg model contains ferromagnetic S^zS^z interactions and a transverse magnetic field.Higher-spin systems are relevant to some nuclear magnetic resonance and condensed-matter applications.
- Boson sampling: Boson sampling is formulated as sampling output occupation distributions from a multimode linear-optical device after injecting N single photons into N of M modes.Its implementation uses ordered effective Hamiltonians for optical elements and first-order Trotterization of each unitary.
VII. FULL COMPOSITE HAMILTONIAN RESULTS
Across five Hamiltonian classes, the optimal encoding depends strongly on the application and truncation or spin size, with four recurring scenarios involving standard binary, Gray, unary, and conversion strategies. The comparisons show that encoding conversions can help in some cases but offer only marginal gains in others.
- Five Hamiltonian classes are compared using relative two-qubit entangling-gate requirements, with bosonic cases extending to d=16 and the Heisenberg model to s=7/2.
- Many plotted SB-and-Gray results improve on SB-only and Gray-only by less than 1%.Conversion and compacting results are omitted when they are no better than the corresponding simpler strategies.